Banach Spaces and Graphs: Geometric Interactions and Applications
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It can arguably be said that the world one lives in is geometric in nature. Numerous practical everyday-life issues, as well as fundamental scientific mysteries, can be expressed in geometric terms. For instance, networks are ubiquitous in our modern society. From the World Wide Web and its powerful search engines to social networks, from telecommunication networks to economic systems, networks represent a wide range of real world systems. They can naturally be seen as geometric objects by considering the number of edges of the shortest path connecting two nodes as a quantity measuring their proximity. The study of physical laws has led as well to the development of a refined mathematical framework where elaborate geometric structures are able to depict and model the interactions of elementary particles and the symmetries underlying quantum physics. The notion of a metric space is a central concept that is pivotal in mathematical models of optimization problems in networks, and in a vast range of application areas, including computer vision, computational biology, machine learning, statistics, and mathematical psychology, to name a few. This extremely useful abstract concept generalizes the classical notion of an Euclidean space, where the distance from point A to point B is computed as the length of an imaginary straight line connecting them. The heart of the matter usually boils down to understanding whether a given metric space, in particular a graph equipped with its shortest path distance, can be faithfully represented in a more structured space, typically a Banach space with some desirable properties. Our ability to do so usually has tremendous applications.The problems investigated in this project are motivated by their potential applications in theoretical physics and theoretical computer science. Most of the problems considered find their origins either in practical issues (e.g. the design of efficient approximation algorithms), or in fundamental mathematical problems in topology or noncommutative geometry (e.g. the Novikov conjecture(s), the Baum-Connes conjecture(s)). Embedding problems that arise in connection to these problems have been considered independently by several groups of mathematicians (Banach space geometers, geometric group theorists, computer scientists...). An underlined aspect of this proposal is to consider these embeddings problems from a unified and global standpoint. Fundamental and long-standing open problems in quantitative metric geometry (e.g. Enflo''s problem, a metric reformulation of uniform smoothability, the coarse embeddability of groups and expander graphs into super-reflexive Banach spaces...) will be tackled from a different angle with new and innovative techniques. In particular, the project will develop a certain asymptotic theory of Banach spaces and explore its connections to the geometry of infinite graphs. The approach here to solve the local problems above, is to study asymptotic counterparts of the local properties involved, in order to gain new insights and to devise new approaches. This approach is motivated by the fact that the asymptotic setting usually provides a finer picture, is on some occasion better understood, and requires completely different techniques. A general outline of the research methodology of this project is to utilize powerful tools from surrounding fields (graph theory, probability theory, Ramsey theory...), and cross-over techniques (e.g. techniques from theoretical computer science to solve geometric group theoretic problems).This award reflects NSF''s statutory mission and has been deemed worthy of support through evaluation using the Foundation''s intellectual merit and broader impacts review criteria.